Optimal designs for indirect regression

Abstract

In many real life applications, it is impossible to observe the feature of interest directly. For example, scientists in Materials Science may be interested in detecting cracks inside objects, not visible from the outside. Similarly, non-invasive medical imaging techniques such as Positrone Emission Tomography rely on indirect observations to reconstruct an image of the patient's internal organs. In this paper, we investigate optimal designs for such indirect regression problems. We determine designs minimizing the integrated mean squared error of estimates of the regression function obtained by Tikhonov or spectral cut-off regularization. We use the optimal designs as benchmarks to investigate the efficiency of the uniform design commonly used in applications. Several examples are discussed to illustrate the results, in most of which the uniform design or a simple modification thereof is demonstrated to be very efficient for the estimation of the regression function. Our designs provide guidelines to scientists regarding the experimental conditions at which the indirect observations should be taken in order to obtain an accurate estimate for the object of interest.